Theory and Design of PID Controller for Nonlinear Uncertain Systems

It is well-known that the classical proportional-integral-derivative (PID) controller plays a fundamental role in various engineering systems. However, up to now a theory that can explain the rationale why the linear PID can effectively deal with nonlinear uncertain dynamical systems and a method that can provide explicit design formula for the PID parameters are still lacking. This motivates our recent study on the theoretical foundation of the PID control. The main purpose of this letter is to extend the 1-D results to higher dimensional nonlinear uncertain systems and to improve the results significantly by a refined method. We will also consider a class of multi-agent uncertain nonlinear systems where each agent is controlled by a PID controller using its own regulation error. We will show that a parameter manifold can be constructed explicitly so that when the PID parameters are chosen from this manifold, the multi-agent systems will be globally stable and the tracking error of each agent will coverage to zero exponentially fast.